Difference between revisions of "1964 AHSME Problems/Problem 32"
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<math> \textbf{(E) }a(b+c+d)=c(a+b+d)</math> | <math> \textbf{(E) }a(b+c+d)=c(a+b+d)</math> | ||
+ | ==Solution== | ||
+ | |||
+ | Cross-multiplying gives: | ||
+ | |||
+ | <math>(a+b)(a+d) = (b+c)(c+d)</math> | ||
+ | |||
+ | <math>a^2 + ad + ab + bd = bc + bd + c^2 + cd</math> | ||
+ | |||
+ | <math>a^2 + ad + ab - bc - c^2 - cd = 0</math> | ||
+ | |||
+ | <math>a(a + b + d) - c(b+c+d)= 0</math> | ||
+ | |||
+ | This looks close to turning into option C, but we don't have a <math>c</math> term in the first parentheses, and we don't have an <math>a</math> term in the second parentheses to allow us to complete the factorization. However, if we both add <math>ac</math> and subtract <math>ac</math> on the LHS, we get: | ||
+ | |||
+ | <math>a(a + b + d) + ac - c(b+c+d) - ca= 0</math> | ||
+ | |||
+ | <math>a(a+b+d +c) - c(b+c+d+a) = 0</math> | ||
+ | |||
+ | <math>(a-c)(a+b+c+d) = 0</math> | ||
+ | |||
+ | This is equivalent to <math>\boxed{\textbf{(C)}</math> | ||
+ | |||
==See Also== | ==See Also== | ||
{{AHSME 40p box|year=1964|num-b=31|num-a=33}} | {{AHSME 40p box|year=1964|num-b=31|num-a=33}} |
Revision as of 01:26, 25 July 2019
Problem
If , then:
Solution
Cross-multiplying gives:
This looks close to turning into option C, but we don't have a term in the first parentheses, and we don't have an term in the second parentheses to allow us to complete the factorization. However, if we both add and subtract on the LHS, we get:
This is equivalent to $\boxed{\textbf{(C)}$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 31 |
Followed by Problem 33 | |
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All AHSME Problems and Solutions |
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